3 THE MODULUS FUNCTION ON A GROUP 265 contradiction, by first showing that the sequence (Ap) may be assumed to have a limit A' <= BO , and then using (c) above to prove that A' contains the frontier S0 of B0 ; finally remark that £S0 + JS0 — B0 .) (e) If A is any compact subset of R", show that An(A)<2-»Vn(S(A)y where the diameter 8(A) is relative to the Euclidean distance, and Vrt is the Lebesgue measure of the ball Bn (*' Bieberbach's inequality")- (Use (d) and the inequality 10. With the notation of Problem 7, for any compact subset A of Rn the upper (resp. lower) Minkowski area of A is defined to be the number a+(A) = lim sup (An(Vr(A)) - An(A))/r r-*0 (resp. or(A) = lim inf (An(Vr(A)) - An(A))/r) r-*0 where r tends to 0 through positive values. (a) Give an example of a compact set A such that a+(A) = 1 and a "(A) = 0. (Take n — 2, and consider a union of finite sets of the form ck -f Afc , where the sequence (cfc) tends to 0, and for each k the set A* is a product Ifc x Jfc, where I*, J* are finite sequences of the form (pak)Q^p^rk, (gbj)o^q^sk. Choose the sequences (cfc), (ak), (bk), (rk), (sk) appropriately.) (b) Show that or(A) ^/ (the isoperimetric inequality). (Use the Brunn-Minkowski inequality.) (c) Let A be a compact convex set (Section 8.5, Problem 8). Show that, for each compact convex set B, the function of £ is convex for 0 <£ £ < + co, by using the Brunn-Minkowski inequality and the relations (a + ]8)A = aA H- 0A, (a 4- ]8)B = aB 4- /?B for all real numbers a, ]8. Deduce that a+(A) = a "(A) (use Problem 8 of Section 8.5). The number a(A) = a+(A) = a~(A) is called the Minkoswki area of A. (d) If the compact convex set A is contained in R"~ *, then a(A) = An_i(A). (e) If 0 is an interior point of a compact convex set A <= R", then wrf1An(A) ^ a(A) ^ nrg1Xn(A)9 where rt is the maximum of the radii of closed balls with center 0 and contained in A, and re is the minimum of the radii of closed balls with 0 which contain A. (Use the inequalities 11. Let G be a compact group, h a left invariant distance on G which defines the topology of G (12.9.1) and JK, a Haar measure on G. Show that is a bi-invariant distance on G, equivalent to h.hat 8(A + B) ^ 8(A) + 8(B).